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1 Tunnel Effect: - particle with kinetic energy E strikes a barrier with height U 0 > E and width L - classically the particle cannot overcome the barrier - quantum mechanically the particle can penetrated the barrier and appear on the other side - then it is said to have tunneled through the barrier examples: - emission of alpha particles from radioactive nuclei by tunneling through the binding potential barrier - tunneling of electrons from one metal to another through an oxide film - tunneling in a more complex systems described by a generalized coordinate varying in some potential phys4.5 Page 1 approximate result: - the transmission coefficient T is the probability of a particle incident from the left (region I) to be tunneling through the barrier (region II) and continue to travel to the right (region III) - depends exponentially on width of barrier L and the difference between the particle kinetic energy and the barrier height (U 0 -E) 1/2 and mass of the particle m 1/2 example: - An electron with kinetic energy E = 1 ev tunnels through a barrier with U 0 = 10 ev and width L = 0.5 nm. What is the transmission probability? - the probability is small, even for a light particle and a thin barrier - but it can be experimentally observed and used in devices phys4.5 Page 2

2 sketch of calculation of tunnel rate: - Schrödinger equation outside of barrier (regions I and III) - has solutions same for ψ III - incoming wave reflected wave - transmitted wave - incoming flux of particles with group velocity v I+ phys4.5 Page 3 transmission: - probability T - ratio of flux of transmitted particles to incident particles barrier region: - Schrödinger equation - solution for U > E - exponentially decaying or increasing wave (no oscillations) - does not describe a moving particle - but probability in barrier region is non-zer0 phys4.5 Page 4

3 boundary conditions: - at left edge of well (x = 0) - at right edge of well (X = L) - solve the four equations for the four coefficients and express them relative to A ( A 2 is proportional to incoming flux) - solution phys4.5 Page 5 transmission coefficient: - find A/F from set of boundary condition equations simplify: - assume barrier U to be high relative to particle energy E simplify: - assume barrier to be wide (k 2 L>1) - therefore phys4.5 Page 6

4 transmission coefficient: - T is exponentially sensitive to width of barrier - T can be measured in terms of a particle flow (e.g. an electrical current) through a tunnel barrier - makes this effect a great tool for measuring barrier thicknesses or distances for example in microscopy applications phys4.5 Page 7 Scanning Tunneling Microscope (STM) xenon atoms on a nickel surface 5 nm moving individual atoms around one by one D.M. Eigler, E.K. Schweizer. Positioning single atoms with a STM. Nature 344, (1990) phys4.5 Page 8

5 Nobel Prize in Physics (1986) "for his fundamental work in electron optics, and for the design of the first electron microscope" "for their design of the scanning tunneling microscope" Ernst Ruska 1/2 of the prize Federal Republic of Germany Fritz-Haber-Institut der Max-Planck- Gesellschaft Berlin, Federal Republic of Germany Gerd Binnig 1/4 of the prize Federal Republic of Germany IBM Zurich Research Laboratory Rüschlikon, Switzerland Heinrich Rohrer 1/4 of the prize Switzerland IBM Zurich Research Laboratory Rüschlikon, Switzerland phys4.5 Page 9 Quantum Harmonic Oscillator general properties: examples: - oscillation around an equilibrium position - at a single frequency - linear restoring force - mechanical oscillator, e.g. mass on a spring - electrical oscillator, e.g. LC-circuit - diatomic molecules - lattice vibrations of a crystal mass on a spring electrical oscillator diatomic molecule phys4.5 Page 10

6 equation of motion: - linear restoring force is a prerequisite for harmonic motion - Hooke's law - equation of motion for harmonic oscillator - a general solution - oscillator frequency note: - in many physical systems the restoring force is not strictly linear in the oscillation coordinate for large amplitude oscillations - for small oscillation amplitudes however, the harmonic oscillator is usually a good approximation - Taylor expansion of any force about the equilibrium position phys4.5 Page 11 potential: - potential associated with Hooke's law - U is used when solving the Schrödinger equation for a harmonic oscillator expectations: - only a discrete set of energies will be allowed for the oscillator - the lowest allowed energy will not be E = o but will have some finite value E = E 0 - there will be a finite probability for the particle to penetrate into the walls of the potential well Schrödinger equation for the harmonic oscillator: phys4.5 Page 12

7 Solving the harmonic oscillator Schrödinger equation: rewrite: normalize: - these are dimensionless units for the coordinate y and the energy α - the Schrödinger equation thus is given by normalization condition for the solution wave functions ψ: phys4.5 Page 13 energy quantization: - condition on α for normalization - energy levels of the harmonic oscillator - equidistant energy levels - this is a distinct feature of the harmonic oscillator - zero point energy (n = 0, lowest possible energy of the harmonic oscillator) phys4.5 Page 14

8 energy levels in different systems: constant potential x 2 potential 1/r - potential particle in a box harmonic oscillator Hydrogen atom phys4.5 Page 15 harmonic oscillator wave functions: - with Hermite polynomials H n - the classical maximum oscillation amplitude is indicated in the plot by vertical black lines - the particle enters into the classically forbidden regions of amplitudes phys4.5 Page 16

9 comparison of classical to quantum probability densities of position classical: quantum: - largest probability density at the turning points (x = ± a) of the oscillation - in the ground state (n = 0) ψ 2 is largest at theequilibrium position (x = 0) - for increasing n the quantum probability density approaches the classical one - n = 10 - the probability for the quantum oscillator to be at amplitudes larger then ± a decreases for increasing n - this is an example of the correspondence principle for large n phys4.5 Page 17 Quantum Harmonic Oscillators Cavity Quantum Electrodynamics (Cavity QED) two photons one photon atom (green) as a source and probe for single photons no photon mirrors (blue) to contain photon in a cavity (a photon box) standing electromagnetic wave with a single photon Review: J. M. Raimond, M. Brune, and S. Haroche Rev. Mod. Phys. 73, 565 (2001) phys4.5 Page 18

10 Cavity QED experimental setup: one result: O: oven as a source of atoms B: LASER preparation stage for atoms C: cavity (photon box) D: atom detector measurement of probability for atom to be in excited state P e versus the time t i spend in cavity atom probes quantum state (number of photons) in the cavity Review: J. M. Raimond, M. Brune, and S. Haroche Rev. Mod. Phys. 73, 565 (2001) phys4.5 Page 19 Quantum HO in Electrical Circuits sketch of electrical circuit: electrical harmonic LC-oscillator inductor L capacitor C electrical artificial atom many nonequidistantly spaced energy levels one photon no photon phys4.5 Page 20

11 Experiment: Quantum HO in a Circuit the LC-oscillator integrated circuit one result: seeing individual photons: artificial atom (blue) LC oscillator (grey) spectrum of artificial atom one line each for 1, 2, 3, photons A. Wallraff, D. Schuster,..., S. Girvin, and R. J. Schoelkopf, Nature (London) 431, 162 (2004) intensity of lines proportional to photon probability phys4.5 Page 21 Photoelectric effect: Intensity two limits: constant frequency different intensities constant intensity?? different frequencies phys4.5 Page 22

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